Engineering Mathematics 1 — Calculus, Matrices, Differential Equations

Unit 1: Calculus — Limits and Continuity

Limits

The limit of a function describes what value f(x) approaches as x approaches a value.

Key Limit Formulas:

lim(x→0) sinx/x = 1
lim(x→0) (1 + x)^(1/x) = e
lim(x→∞) (1 + 1/x)^x = e
lim(x→0) (eˣ - 1)/x = 1
lim(x→0) (aˣ - 1)/x = ln(a)

L'Hôpital's Rule: If limit gives 0/0 or ∞/∞ form, differentiate numerator and denominator separately.

Differentiation Formulas (Must Memorize)

| Function | Derivative | |---|---| | xⁿ | nxⁿ⁻¹ | | eˣ | eˣ | | aˣ | aˣ ln(a) | | ln(x) | 1/x | | sin(x) | cos(x) | | cos(x) | -sin(x) | | tan(x) | sec²(x) | | sin⁻¹(x) | 1/√(1-x²) | | tan⁻¹(x) | 1/(1+x²) |

Chain Rule: d/dx[f(g(x))] = f'(g(x)) · g'(x)

Product Rule: d/dx[u·v] = u'v + uv'

Quotient Rule: d/dx[u/v] = (u'v - uv') / v²

Applications of Derivatives

• Maxima/Minima: f'(x)=0, then check f''(x); if f''<0 → max, f''> 0 → min
• Increasing/Decreasing: f'(x)>0 → increasing; f'(x)<0 → decreasing
• Rolle's Theorem: If f(a)=f(b), then f'(c)=0 for some c in (a,b)
• Mean Value Theorem: f'(c) = [f(b)-f(a)]/(b-a) for some c in (a,b)

Integration Formulas

| Function | Integral | |---|---| | xⁿ | xⁿ⁺¹/(n+1) + C | | 1/x | ln|x| + C | | eˣ | eˣ + C | | sin(x) | -cos(x) + C | | cos(x) | sin(x) + C | | sec²(x) | tan(x) + C |

Integration by Parts: ∫u dv = uv − ∫v du ILATE Rule: Choose u in order: Inverse trig, Logarithm, Algebraic, Trigonometric, Exponential

Definite Integral: ∫ₐᵇ f(x)dx = F(b) - F(a)

---

Unit 2: Matrices and Determinants

Matrix Operations

A matrix is a rectangular array of numbers.

Types: Row matrix (1×n), Column matrix (m×1), Square matrix (n×n), Identity matrix (I), Zero matrix

Matrix Multiplication:

• A(m×n) × B(n×p) = C(m×p)
• Number of columns of A must equal number of rows of B
• Not commutative: AB ≠ BA generally

Determinants

2×2 Determinant:

|a b|
|c d| = ad - bc

3×3 Determinant (Sarrus Rule or Cofactor Expansion):

|a b c|
|d e f| = a(ei-fh) - b(di-fg) + c(dh-eg)
|g h i|

Properties of Determinants:

• |AB| = |A|·|B|
• |Aᵀ| = |A|
• If two rows are identical → determinant = 0
• Swapping two rows changes sign of determinant

Cramer's Rule (Solving Linear Equations)

For system AX = B: x₁ = D₁/D, x₂ = D₂/D, x₃ = D₃/D Where D = det(A), Dᵢ = det(A with i-th column replaced by B)

Matrix Inverse

A⁻¹ = (1/|A|) × adj(A)

Conditions: Inverse exists only if |A| ≠ 0 (non-singular matrix)

Rank of a Matrix

Number of non-zero rows in Row Echelon Form.

Consistency of equations (Rouché-Capelli):

• rank(A) = rank(augmented) = n → unique solution
• rank(A) = rank(augmented) < n → infinite solutions
• rank(A) ≠ rank(augmented) → no solution

---

Unit 3: Differential Equations

A differential equation relates a function with its derivatives.

Order: Highest derivative present (d²y/dx² → order 2) Degree: Power of highest order derivative

First Order ODEs

Separable: dy/dx = f(x)g(y) → separate and integrate both sides

Linear ODE: dy/dx + P(x)y = Q(x)

• Integrating Factor: IF = e^∫P dx
• Solution: y × IF = ∫(Q × IF)dx

Exact ODE: M dx + N dy = 0 where ∂M/∂y = ∂N/∂x

Second Order Linear ODE with Constant Coefficients

ay'' + by' + cy = 0

Characteristic Equation: am² + bm + c = 0

| Roots | Solution | |---|---| | Two real distinct m₁, m₂ | y = C₁e^(m₁x) + C₂e^(m₂x) | | Equal roots m₁ = m₂ = m | y = (C₁ + C₂x)e^(mx) | | Complex α ± βi | y = e^(αx)(C₁cos βx + C₂sin βx) |

---

Solved PYQ Questions

Q1 (2023): Find dy/dx if y = x³ sin(x) Using Product Rule: dy/dx = 3x² sin(x) + x³ cos(x)

Q2 (2023): Evaluate ∫₀¹ x·eˣ dx Using Integration by Parts (u=x, dv=eˣdx): = [x·eˣ]₀¹ - ∫₀¹ eˣ dx = e - [eˣ]₀¹ = e - (e-1) = 1

Q3 (2022): Find inverse of A = [[2,1],[5,3]] |A| = 2×3 - 1×5 = 6-5 = 1 adj(A) = [[3,-1],[-5,2]] A⁻¹ = (1/1)×[[3,-1],[-5,2]] = [[3,-1],[-5,2]]

Q4 (2022): Solve dy/dx + y = eˣ Linear ODE: P=1, Q=eˣ IF = e^∫1dx = eˣ y·eˣ = ∫eˣ·eˣ dx = ∫e²ˣ dx = e²ˣ/2 + C y = eˣ/2 + Ce⁻ˣ

---

Quick Revision Checklist

• Differentiation formulas: xⁿ, eˣ, sinx, cosx, lnx, tan⁻¹x
• Chain rule, product rule, quotient rule
• Integration formulas and ILATE rule
• 2×2 and 3×3 determinant calculation
• Cramer's rule steps
• Matrix inverse: A⁻¹ = adj(A)/|A|
• First order ODE: separable and linear types
• Second order ODE: characteristic equation and 3 cases of roots